Equation of Motion Series Expansion of Double Time Green's Functions

TL;DR

This paper introduces a method to expand double time Green's functions using Taylor series in a parameter, employing continued fractions for resummation, and applies it to the Anderson impurity model to improve local density of states calculations.

Contribution

It develops a novel series expansion and resummation technique for Green's functions, addressing divergence issues and enhancing accuracy in impurity model analysis.

Findings

Effective series expansion for Green's functions demonstrated

Resummation via continued fractions improves analytical structure

Enhanced local density of states results in Anderson model

Abstract

Based on the Green's function (GF) equation-of-motion formalism, we develop a method to expand the double time Green's function into Taylor series of the parameter $\lambda$ in the Hamiltonian $H=H_0 + \lambda H_1$. Here $H_0$ is the exactly solvable part and $H_1$ is regarded as the perturbation. To restore the analytical structure of GF, we use the continued fraction to do resummation for the obtained series. The problem of zero-temperature divergence is identified and remedied by the self-consistent series expansion. To demonstrate the implementation of this method, we carry out the weak- as well as the strong-coupling expansion for the Anderson impurity model to order $\lambda^2$. Improved result for the local density of states is obtained by self-consistent second-order strong-coupling expansion and continued fraction resummation.